Optimization with PDE Constraints

By: Michael Hinze, Rene Pinnau, Michael Ulbrich

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Hardcover | 14 November 2008

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284 Pages

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Solving optimization problems subject to constraints given in terms of partial d- ferential equations (PDEs) with additional constraints on the controls and/or states is one of the most challenging problems in the context of industrial, medical and economical applications, where the transition from model-based numerical si- lations to model-based design and optimal control is crucial. For the treatment of such optimization problems the interaction of optimization techniques and num- ical simulation plays a central role. After proper discretization, the number of op- 3 10 timization variables varies between 10 and 10 . It is only very recently that the enormous advances in computing power have made it possible to attack problems of this size. However, in order to accomplish this task it is crucial to utilize and f- ther explore the speci?c mathematical structure of optimization problems with PDE constraints, and to develop new mathematical approaches concerning mathematical analysis, structure exploiting algorithms, and discretization, with a special focus on prototype applications. The present book provides a modern introduction to the rapidly developing ma- ematical ?eld of optimization with PDE constraints. The ?rst chapter introduces to the analytical background and optimality theory for optimization problems with PDEs. Optimization problems with PDE-constraints are posed in in?nite dim- sional spaces. Therefore, functional analytic techniques, function space theory, as well as existence- and uniqueness results for the underlying PDE are essential to study the existence of optimal solutions and to derive optimality conditions.

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"The book presents a state-of-the-art of optimization problems described by partial differential equations (PDEs) and algorithms for obtaining their solutions. Solving optimization problems with constraints given in terms of PDEs is one of the most challenging problems appearing, e.g., in industry, medical and economical applications. The book consists of four chapters. ... This well-written book can be recommended to scientists and graduate students working in the fields of optimal control theory, optimization algorithms and numerical solving of optimization problems described by PDEs." (Wieslaw Kotarski, Zentralblatt MATH, Vol. 1167, 2009)

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You Can Find This Book In

Non-Fiction Mathematics Optimisation Linear Programming Calculus & Mathematical Analysis Calculus of Variations

Preface	p. v
Analytical Background and Optimality Theory	p. 1
Stefan Ulbrich	p. 1
Introduction and Examples	p. 1
Introduction	p. 1
Examples for Optimization Problems with PDEs	p. 4
Optimization of a Stationary Heating Process	p. 5
Optimization of an Unsteady Heating Processes	p. 7
Optimal Design	p. 8
Linear Functional Analysis and Sobolev Spaces	p. 9
Banach and Hilbert Spaces	p. 10
Sobolev Spaces	p. 13
Weak Convergence	p. 24
Weak Solutions of Elliptic and Parabolic PDEs	p. 26
Weak Solutions of Elliptic PDEs	p. 26
Weak Solutions of Parabolic PDEs	p. 36
Gateaux- and Fréchet Differentiability	p. 50
Basic Definitions	p. 50
Implicit Function Theorem	p. 52
Existence of Optimal Controls	p. 52
Existence Result for a General Linear-Quadratic Problem	p. 52
Existence Result for Nonlinear Problems	p. 54
Applications	p. 56
Reduced Problem, Sensitivities and Adjoints	p. 57
Sensitivity Approach	p. 58
Adjoint Approach	p. 59
Application to a Linear-Quadratic Optimal Control Problem	p. 60
A Lagrangian-Based View of the Adjoint Approach	p. 63
Second Derivatives	p. 64
Optimality Conditions	p. 65
Optimality Conditions for Simply Constrained Problems	p. 65
Optimality Conditions for Control-Constrained Problems	p. 70
Optimality Conditions for Problems with General Constraints	p. 80
Optimal Control of Instationary Incompressible Navier-Stokes Flow	p. 88
Functional Analytic Setting	p. 89
Analysis of the Flow Control Problem	p. 91
Reduced Optimal Control Problem	p. 94
Optimization Methods in Banach Spaces	p. 97
Michael Ulbrich	p. 97
Synopsis	p. 97
Globally Convergent Methods in Banach Spaces	p. 99
Unconstrained Optimization	p. 99
Optimization on Closed Convex Sets	p. 104
General Optimization Problems	p. 109
Newton-Based Methods-A Preview	p. 109
Unconstrained Problems-Newton's Method	p. 109
Simple Constraints	p. 110
General Inequality Constraints	p. 113
Generalized Newton Methods	p. 115
Motivation: Application to Optimal Control	p. 115
A General Superlinear Convergence Result	p. 116
The Classical Newton's Method	p. 119
Generalized Differential and Semismoothness	p. 120
Semismooth Newton Methods	p. 123
Semismooth Newton Methods in Function Spaces	p. 125
Pointwise Bound Constraints in L²⁺	p. 125
Semismoothness of Superposition Operators	p. 126
Pointwise Bound Constraints in L²⁺ Revisited	p. 129
Application to Optimal Control	p. 130
General Optimization Problems with Inequality Constraints in L²⁺	p. 132
Application to Elliptic Optimal Control Problems	p. 133
Optimal Control of the Incompressible Navier-Stokes Equations	p. 137
Sequential Quadratic Programming	p. 140
Lagrange-Newton Methods for Equality Constrained Problems	p. 140
The Josephy-Newton Method	p. 144
SQP Methods for Inequality Constrained Problems	p. 148
State-Constrained Problems	p. 151
SQP Methods	p. 152
Semismooth Newton Methods	p. 152
Further Aspects	p. 155
Mesh Independence	p. 155
Application of Fast Solvers	p. 156
Other Methods	p. 156
Discrete Concepts in PDE Constrained Optimization	p. 157
Michael Hinze	p. 157
Introduction	p. 157
Control Constraints	p. 158
Stationary Model Problem	p. 158
First Discretize, Then Optimize	p. 160
First Optimize, Then Discretize	p. 161
Discussion and Implications	p. 163
The Variational Discretization Concept	p. 164
Error Estimates	p. 167
Boundary Control	p. 177
Some Literature Related to Control Constraints	p. 196
Constraints on the State	p. 197
Pointwise Bounds on the State	p. 198
Pointwise Bounds on the Gradient of the State	p. 219
Time Dependent Problem	p. 227
Mathematical Model, State Equation	p. 227
Optimization Problem	p. 229
Discretization	p. 229
Further Literature on Control of Time-Dependent Problems	p. 231
Applications	p. 233
Rene Pinnau	p. 233
Optimal Semiconductor Design	p. 233
Semiconductor Device Physics	p. 234
The Optimization Problem	p. 240
Numerical Results	p. 246
Optimal Control of Glass Cooling	p. 250
Modeling	p. 251
Optimal Boundary Control	p. 254
Numerical Results	p. 260
References	p. 265
Table of Contents provided by Ingram. All Rights Reserved.

Optimization with PDE Constraints

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